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I want to prove the existence and unicity of a minimiser of the energy functional $u \to E(u) = \int_\Omega |\nabla u|^2$ for $\Omega$ a smooth domain in $\mathbb{R}^n$ with $C^1$ boundary and $u \in U := \{ u \in H^1(\Omega) : u|_{\partial \Omega} = g \}, $ for some smooth $g$.

My strategy is the following. Because the energy functional is strictly convex, unicity is guaranteed (if there were two different minimizers $u,v$, then $\frac{u+v}{2}$ would have strictly lower energy). Now take a minimizing sequence $(u_k)$ ; it is bounded in $H^1$ (simply because the norm in $H^1$ is $E(u)+||u||_{L^2}$), and $U$ is a Hilbert space (as a closed subset of $H^1$, which is Hilbert), so $U$ is reflexive and thus I can extract a weakly converging subsequence $u_k \xrightarrow{\text{weakly}}{u} \in U$. Now I just have to prove weak lower-semicontinuity of $E$.

This is where I got stuck. I tried $\int_\Omega \nabla u \cdot \nabla u = \lim_k \int_\Omega \nabla u \cdot (\nabla u - \nabla u_k) + \nabla u_k \cdot \nabla u_k$, but I don't see why $ \int_\Omega \nabla u \cdot (\nabla u - \nabla u_k)$ should necessarily converge to 0.

I have read that this might be linked to the convexity of $E$, but I am looking for a self-contained answer to fully understand. Many thanks :)

Suspicious Fred
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  • You did not use the boundary data for unicity, i.e. in the general case this energy is not strictly convex. E.g. $u=5$, $v=1$, i.e. two different constant function, would yield $E(u)=E(v)=0$ and since $E\geq 0$ you would have two different minimiser. – humanStampedist Jun 22 '23 at 11:39
  • Another nitpick: $U$ is not a vector space, since if $g\neq0$, then $0\notin U$. Also you need Poincares inequality to show, that the $H^1$-norm is bounded, because otherwise $|u|_{L^2}$ may not be bounded. To your original question: You may use the definition of weak convergence and define a suitable linear continuous function. – humanStampedist Jun 22 '23 at 11:45
  • It is weakly lower semicontinuous because it is convex and continuous. – daw Jun 22 '23 at 12:50

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